Inverse Problems
The study of inverse problems is a well-established discipline with several applications in a variety of fields, including signal processing and synthetic aperture radar (SAR) systems. The theory of inverse problems examines when and how a function can be inverted to infer the input from the output of the function. Such problems are abundant in signal acquisition and processing systems. The formulation of these problems as inverse problems has produced significant results in the areas of image de-noising, de-blurring and super-resolution.
A common inverse problem in signal processing is the recovery of a signal x from a set of measurements y,y=A(x)+n,  (1)where the function A models the acquisition system, and n represents noise. The goal of the reconstruction process is to determine a signal estimate {circumflex over (x)} that is close to the signal x.
If the acquisition function A is invertible and the noise is negligible, an obvious choice is to use the inverse of the function A to determine x. However, that method can fail if A is not injective, i.e., the data can be explained by multiple signals, or if there is substantial noise. A more general approach is to estimate x by minimizing the following cost function
                                          x            ⋒                    =                                    arg              ⁢                                                          ⁢                                                min                  x                                ⁢                                  f                  ⁡                                      (                                          y                      ,                                              A                        ⁡                                                  (                          x                          )                                                                                      )                                                                        +                          λ              ⁢                                                          ⁢                              g                ⁡                                  (                  x                  )                                                                    ,                            (        2        )            where ƒ(•, •) is a cost function measuring data fidelity according to the properties of the acquisition system and the noise, g(x) is a regularizer that incorporates knowledge about the properties of the signal of interest and penalizes unwanted solutions, and λ controls the trade-off between the two terms in the cost function. The formulation in Eqn. (2) offers flexibility in accommodating a range of acquisition scenarios and signal models.
In the special case where ƒ(y,A(x))=∥y−A(x)∥2, g(x)=0, the function A is linear, and the system is overdetermined, the solution to Eqn. (2) uses a pseudoinverse function A†. The pseudoinverse is also the solution when the system is underdetermined and g(x)=∥x∥2, i.e., a least-energy solution is desired.
SAR Image Formation
The formation of SAR images can be formulated as an inverse problem. Specifically, the ideal SAR acquisition process can be viewed as a linear system, i.e., an instance of Eqn. (1), wherein x represents a two-dimensional (2-D) image of surface reflectivity, y represents the received signal, and the function A is linear.
Henceforth, A is used to represent a linear acquisition function. The received signal y is usually arranged in a 2-D form similar to the SAR image. Samples of each reflected pulse can be thought of as forming a row vector, with different reflected pulses stacked together to form a matrix of data. To emphasize that the function A is linear, we use the notation Ax to denote the application of function A(•) on the ground signal (image) x. Ax can be considered as a matrix multiplication when we rearrange x in one single column vector instead of a 2D image.
Each row of the data matrix y corresponds to a different position of the acquisition platform along its path. The dimension along the path is referred to as the azimuth. Each column corresponds to a delay from the transmission of a pulse. Because the delay is proportional to the distance the pulse has traveled, this second dimension is referred to as the range.
Several known image formation procedures can be interpreted from the viewpoint of inverse problems as determining approximations to the pseudoinverse function A†. The approximations enable efficient implementations to estimate the SAR image x. For example, many procedures rely on pulse compression, which refers to the approximate deconvolution of the received signal through correlation with the transmitted pulse. The correlation can be made highly efficient by using a Fast Fourier Transform (FFT). Pulse compression corresponds to the exact inverse, at least within the bandwidth of interest, if the received pulse is sampled at the Nyquist rate, and corresponds closely to the pseudoinverse if the received pulse is slightly oversampled, which is usually the case in practical SAR systems.
One of the most widely used image formation procedures is the Chirp Scaling Algorithm (CSA), Runge et al., “A novel high precision SAR focusing algorithm based on chirp scaling,” International Geoscience and Remote Sensing Symposium (IGARSS), May 1992, pp. 372-375, incorporated herein by reference.
As shown in FIG. 1, an azimuth FFT 110 is applied to an input signal 101, followed by chirp scaling 120, a range FFT 130, range compression 140, a range IFFT 150, chirp scaling 160, azimuth compression 170, and an azimuth inverse FFT (IFFT) 180 to produce an output signal 109.
The chirp scaling approximates a time-varying delay of a linear chirp signal by means of multiplications with two other chirp signals. Chirp scaling is used in the CSA to correct for range migration, the variation in distance to a given target, and hence an associated delay, caused by the motion of the acquisition platform. The CSA is very efficient due to its composition in terms of FFTs and multiplications only. For this reason, CSA forms the basis for our method of determining the acquisition function and its adjoint as described below.
Saturation and Inverse Problems
Saturation of data values is a significant problem in SAR signal acquisition because of large fluctuations in data amplitudes and the use of low-precision quantizers. Conventional image formation methods, such as the CSA, usually do not address data saturation specifically.
Signal saturation is a very common problem encountered in analog-to-digital (A/D) conversion systems because electronic components have a finite voltage range beyond which the signal amplitude is not allowed to vary. In addition, quantizers use a finite number of quantization levels. If the acquired signal amplitude varies beyond saturation threshold values ±T, then the acquired amplitude saturates to ±T.
Although saturation on its own is undesirable, in the presence of severe quantization, saturation can be beneficial. Specifically, increasing the gain of the signal and promoting saturation also increases the signal-to-quantization noise ratio in the unsaturated measurements. If the saturation is appropriately taken into account, overall reconstruction performance can be increased.
The most common approach to handling saturation, i.e., using the saturated values at face value in the reconstruction as if no saturation occurred, produces severe artifacts. The reconstruction error can be reduced using a consistent reconstruction approach, i.e., ensuring that the reconstructed signal estimate produces the same saturation if re-acquired. Consistent reconstruction significantly improves the reconstruction error when used with quantized and saturated measurements.